\newproblem{lay:1_1_33}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.1.33}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin-plate when the temperature
	around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the
	direction perpendicular to the plate. Let $T_1$, ..., $T_4$ denote the temperatures at the four interior nodes of the mesh in the figure. The
	temperature at a node is approximately equal to the average of the four nearest nodes (to the left, below, right and above). For instance,
	\begin{center}
		$T_1=\frac{1}{4}(10+20+T_2+T_4)$\\
		\includegraphics[scale=0.5]{Tema2/lay_1_1_33.eps}
	\end{center}
	Write a system of four equations whose solution gives estimates for the temperatures $T_1$, ..., $T_4$
}{
   % Solution
	The following equations express the temperatures at each node as the average of the four surrounding nodes.
	\begin{center}
		$T_1=\frac{1}{4}(10+20+T_2+T_4)$\\
		$T_2=\frac{1}{4}(20+40+T_1+T_3)$\\
		$T_3=\frac{1}{4}(30+40+T_2+T_4)$\\
		$T_4=\frac{1}{4}(10+30+T_1+T_3)$\\
	\end{center}
	We may rewrite this equation system as
	\begin{center}
	  $\begin{array}{rrrrcl}
		 T_1            &-\frac{1}{4}T_2&               &-\frac{1}{4}T_4&=&7.5\\
		 -\frac{1}{4}T_1&           +T_2&-\frac{1}{4}T_3&               &=&15\\
		                &-\frac{1}{4}T_2&           +T_3&-\frac{1}{4}T_4&=&17.5\\
		 -\frac{1}{4}T_1&               &-\frac{1}{4}T_3&           +T_4&=&10\end{array}$
	\end{center}
}
\useproblem{lay:1_1_33}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
